A Nonlinear Age-Structured Model of Population Dynamics with Inherited Properties

Abstract

In this paper, we present some results regarding existence and uniqueness of solution on L p -spaces, 1 < p < +∞, to a nonlinear initial boundary value problem originally proposed by Lebowitz and Rubinow (J Math Biol 1:17-36, 1974) to model an age-structured cell population with inherited properties. Our results complete those obtained by Garcia

An existence and uniqueness principle for a nonlinear version of the Lebowitz‐Rubinow model with infinite maximum cycle length

“…Hence, at mitosis, daughter and mother cells are related by a nonlinear rule, which describes the boundary conditions. It writes in the shape f ( t ,0, l )=[ R f ( t ,·,·)]( l ), where R denotes a nonlinear operator on suitable trace spaces and is intended to model the transition from mother cells with cycle length l to daughter cells with cycle length l …”

“…Thus, we can consider the following nonlinear problem: We point out that in García‐Falset the well‐posedness of problem was discussed in the space L 1 when the maximum cycle length is finite for local boundary conditions. Later, in Al‐Izeri and Latrach, they completed the above study showing that problem , when , has a unique mild solution in L p ‐spaces with . The case for nonlocal boundary conditions on L p ‐spaces was considered in Shanthidevi et al A stationary version of this problem was also discussed in Latrach et al However, it seems that the well‐posedness of problem when the maximum cycle length has not yet been investigated.…”